4x4 centers

My record average for solving the 4x4 cube is 1 minute 20.78 seconds, that's 19.57 seconds slower than the current world record (see 4x4 at SCC). Most of us first solve the centers, then pair up the edges, then solve like a 3x3. Compared to the fastest guys I lose time in the latter two steps, but my centers solving is fairly fast with a record average of 13.82 seconds (see 4x4 centers at SCC). I came up with my own method for this, and the people I showed it so far liked it. It's quite easy yet very effective. You could call it a "layer-by-layer" method:

Solve white center

With white on the left side, solve the l-slice

With white on the bottom side, solve the u-slice

This will of course automatically solve the yellow center as well. Note that sometimes with "center" I mean a single piece (of the 24), other times I mean the set of four pieces of the same color, but it should always be clear from context what I mean.

Maybe I'll add applets and a better video later, for now have a look at this slow demo video where I somewhat try to point out what I'm doing. I suggest to watch it now just to see it in action, and watch it again after reading the description below.

Step 1: Solve white center

This should take about 5 moves. The idea is to build two pairs of adjacent white centers on separate faces and then match the two pairs. Usually it goes like this:

Build a pair. Sometimes there already is one, most of the time though it takes one move.

Adjust a third center so that it can be connected with the fourth in one more move. Sometimes I do this before step 1.

Build the second pair.

Adjust a pair so that it can be connected with the other in one more move.

Connect the two pairs.

Once you decide on the first two steps, the last three are just straightforward. But you often have choices for the first two steps and you can combine or reverse them. So use your preinspection time to consider several possibilities and choose the overall best one. Take 5 moves as a goal. I just did a 4.4 move average-of-10, using Jaap's scrambler. Single results: 5 4 5 5 (6) 4 5 4 (3) 4 5 3.

Step 2: White on L, solve the l-slice

Rotate the cube so that the white center is on the left side. Now we solve the l-slice, using only U, (Ll) and (Rr) turns and x-rotations of the cube. Ok, sometimes maybe also an F turn. You have to know the relative order of the four colors for this. The idea here is to build pairs and put them into the l-slice where they belong. Very often you'll already have one piece already solved and you build the pair with an (Rr) turn and then finish the pair into the l-slice with U or U'.

Step 3: White on D, solve the u-slice

Rotate the cube with z' so that the white center is on the bottom side. Now we solve the u-slice. The idea is to "shoot down" center pieces from U into their correct face (determined by the d-slice). Each shot is either (Rr)' F (Rr) or (Ll) F' (Ll)'. If there's a center of your current F-color on U, then shoot it down into F. If not, rotate the cube so that one of the colors currently on U becomes your F face.

Improvements

Some of these have been influenced or directly suggested by fellow cubers, thanks a lot guys!

Instead of (Rr)' F (Rr) and (Ll) F' (Ll)' you can of course also use (Ll)' U (Ll) and (Rr) U' (Rr)', respectively. That's the same, just holding/rotating the cube as whole a bit differently. Actually I think I'm somewhere in the middle, tilting the cube about 45 degrees, i.e. doing half (Rr) and half (Ll). Do what fits your hands best.

At the end of step 2, turn (Rr) to solve one or more centers in the r-slice (i.e. match with the l-slice) to get a shorter step 3. In the above video I did (Rr)', though (Rr)2 would've been smarter because that would've solved 2 centers instead of 1 (you can see a green center on the blue face and a red one on the orange face, that screams for (Rr)2).

For step 1, also consider starting with yellow instead of white if it's easier. Then you have to reverse the order of the four colors in step 2 (and 3) but that's easy to get used to. I tried this with the same scrambles I used for the above average and got a 4.2 moves average for solving the yellow center, single results were (3) (5) 5 5 4 3 5 4 5 3 5 3. Had I combined that, i.e. choose the better of white and yellow as start, it would be a 3.9 move average for solving the first center!

If the first center is easy, look whether you can preserve or build pairs for step 2 during step 1.

During step 3, to shoot to the B-face you don't need to do a y2 cube rotation. Instead simply use (Rr)' U (Rr) or (Ll) U' (Ll)'. So usually I solve all centers on U that belong to my current F/B, and only when there are none left, rotate the cube with y or y'.

Shooting down a connected pair is very smooth, the two shots combine nicely as (Rr)' F (RrLl) F' (Ll)'. You just need to turn them to Ufl and Ufr first.

A diagonally separated pair on U will automatically become connected when you shoot away one of the other two first.

Another (direct) way to solve such a separated pair is simply shooting them down separately with U or U' between the two shots. I use this if I end up with a separated pair, or if I have to shoot it to F/B and the intervening shot (that would connect them) goes to L/R and thus requires a cube rotation. An advantage of the separated pair is that you don't need setup moves, you can shoot it down right away.

Shooting a pair (connected or separated) to B doesn't even require the quick glance at B that a single shot to B does.

Use your index fingers to pull, not your thumbs to push, with the exception of the rare F' during step 2.

Look ahead and go smooth. Just like 3x3 F2L, don't go full speed but rather a little slower. You can't turn a 4x4 as fast as a 3x3 anyway and turning it "smoothly" is a general advice for it.

Step 3: prevent solving the yellow center early on, before having solved the others. That's like running out of ammo. Whenever you shoot one piece out of U, a new one gets into U. If you have a choice (shooting left or right), prefer to get a non-yellow piece to a yellow one to get into U.

If you do solve yellow before all others are solved, usually only two or three centers are unsolved. Rotate the cube (usually z/z') so that one of them gets on U and a solved one gets on D. Then continue step 3, just with a different top-color than yellow.

In rare cases you can't complete the l-slice during step 2, because there are less than two centers of one color in the l and r slices together. That's because three or more of that color are on the R face (opposite white). This means it can be only one color. So after solving the l-slice in step 2 as far as you can, turn (Rr) as suggested before to get a bit for free, and then continue to step 3. I'd first solve the color that has 3 (or 4) centers on top now, using something like (Rr)' F2 (RrLl) F' (Ll)' or (Rr)' F2 (RrLl) F2 (Ll)'.

I haven't thoroughly tested this, but it feels better when I don't hold my elbows close to my body but about 15 centimeters to the sides. The cube stays at the same position, just "raise" your elbows. This changes the angle of your arms and the fingers.

During step 3, instead of for example y U' you can do ud'D'.Another case of "do what works best for you".

There are a lot of choices throughout, not only in the beginning of step 1. For step 2, I usually solve first whatever I see first, particularly I don't necessarily walk around the cube (like I did in the video) but might jump to the opposite side for the next pair if that's what I see. Or I might walk around the other direction (x instead of x'), which actually sounds smarter now that I think about it. For step 3, I also often don't shoot to B if I can, but instead rotate y or y' and continue there. It depends a bit on what I see and also where the pieces are on U. You can also save some U-adjustments there. So here's a very general advice: don't follow my description strictly. If you want to use my method, take my description only as a general guide, but find out what works best for you. And try to find your own further improvements.